The proof is decomposed into several steps.
Step 1. We first prove \((\Delta ^2+E)(H+E)^{-1}\in \mathbb {B}(L^p)\) .
Since \(H\)  is bounded below, there exists \(E_0>0\)  such that if \(E\ge E_0\)  then \(H+E\)  is a positive self-adjoint operator and
\((H+E)^{-1}f=\int _0^\infty e^{-tH}e^{-Et}fdt,\quad f\in L^2.\) 
It was proved by Deng–Ding–Yao [1]} that \(e^{-tH}\)  (initially defined on \(L^2\) ) extends to an analytic semi-group \(e^{-zH}\)  on \(L^1\)  with angle \(\pi /2\)  and its kernel satisfies:
\(|e^{-tH}(x,y)|\lesssim t^{-1/4}\exp \left(-\frac{c|x-y|^{4/3}}{t^{1/3}}+\omega t\right),\quad t>0\) 

with some constant \(c,\omega >0\) . In particular, \(e^{-tH}\in \mathbb {B}(L^p({\mathbb {R}}))\)  for all \(1\le p\le \infty ,t\ge 0\)  and
\({\Vert e^{-tH}\Vert }_{L^p\rightarrow L^p}\lesssim e^{\omega t}.\) 
In what follows, we alway assume \(E>\max (E_0,\omega )\) . Then, for \(f\in L^2\cap L^p\) ,
\({\Vert (H+E)^{-1}f\Vert }_{L^p}\le \int _0^\infty e^{-Et}{\Vert e^{-tH}f\Vert }_{L^p}dt\lesssim \int _0^\infty e^{-(E-\omega )t}dt{\Vert f\Vert }_{L^p}\lesssim |E-\omega |^{-1}{\Vert f\Vert }_{L^p}.\) 
Hence \((H+E)^{-1}\)  extends to a bounded operator on \(L^p\) . Moreover, we have
\(\Delta ^2(H+E)^{-1}f=(H+E-V-E)(H+E)^{-1}f=1-(V+E)(H+E)^{-1}f\) 
and hence \({\Vert \Delta ^2(H+E)^{-1}f\Vert }_{L^p}\lesssim (1+{\Vert V\Vert }_{L^\infty }){\Vert f\Vert }_{L^p}\)  for all \(f\in L^2\cap L^p\) .
By the density argument, we thus obtain \((\Delta ^2+E)(H+E)^{-1}\in \mathbb {B}(L^p)\) .
Step 2. Next we prove \((\Delta ^2+E)^{s/4}(H+E)^{-s/4}\in \mathbb {B}(L^p)\)  for \(0<s<4\) . It follows from [lemma62proof1]lemma62proof1 that \(H+E\)  satisfies the generalized gaussian bound:
\(|e^{-t(H+E)}(x,y)|\lesssim t^{-1/4}\exp \left(-\frac{c|x-y|^{4/3}}{t^{1/3}}\right),\quad t>0,\quad E>\max (E_0,\omega ).\) 

With this bound at hand, we can apply the abstract spectral multiplier theorem by Blunck [2]} to \(H+E\)  obtaining
\({\Vert (H+E)^{i\beta }\Vert }_{L^p\rightarrow L^p}\le C_p{\langle }\beta {\rangle }^2,\quad 1<p<\infty ,\ \beta \in {\mathbb {R}}.\) 
This \(L^p\) -bounds allow us to interpolate between the trivial case \(s=0\)  and the case \(s=4\)  proved in the above Step 1 by applying Stein's analytic interpolation theorem [3]}, yielding \((\Delta ^2+E)^{s/4}(H+E)^{-s/4}\in \mathbb {B}(L^p)\)  for \(0<s<4\) .
Step 3. Next, we prove by induction that \((\Delta ^2+E)^{-N-1}(H+E)^{-N-1}\in \mathbb {B}(L^p)\)  if \(V\in B^{4N}({\mathbb {R}})\) . The case \(N=0\)  holds by Step 1. If \(N\ge 1\) , we find by the resolvent equation that
\((H+E)^{-N-1}f=(\Delta ^2+E)^{-1}(H+E)^{-N}f-(\Delta ^2+E)^{-1}V(H+E)^{-N-1}f,\quad f\in L^2.\) 
We also know that \((H+E)^{-N},V(H+E)^{-N-1}\in \mathbb {B}(L^p,W^{4N,p})\)  by the assumption on \(V\) , the fact \((H+E)^{-1}\in \mathbb {B}(L^p)\)  and the induction hypothesis. Moreover, it is well known that \((\Delta ^2+E)^{-1}\in \mathbb {B}(W^{4(N-1),p},W^{4N,p})\) . Therefore, it follows for \(f\in L^p\cap L^2\)  that
\({\Vert (H+E)^{-N-1}f\Vert }_{W^{4N,p}}\lesssim {\Vert (H+E)^{-N}f\Vert }_{W^{4N,p}}+{\Vert V(H+E)^{-N-1}f\Vert }_{W^{4N,p}}\lesssim {\Vert f\Vert }_{L^p}\) 

Hence \((\Delta ^2+E)^{-N-1}(H+E)^{-N-1}\in \mathbb {B}(L^p,W^{4(N+1),p})\)  by the density argument.
Step 4. The same interpolation argument as above with \((H+E)^{-1}\)  replaced by \((H+E)^{-N-1}\) , together with the above Step 3, implies \((\Delta ^2+E)^{s/4}(H+E)^{-s/4}\in \mathbb {B}(L^p)\)  for all \(0<s<4(N+1)\)  if \(V\in B^{4N}({\mathbb {R}})\) . This completes the proof of \((\Delta ^2+E)^{s/4}(H+E)^{-s/4}\in \mathbb {B}(L^p)\) . The proof of \((H+E)^{s/4}(\Delta ^2+E)^{-s/4}\in \mathbb {B}(L^p)\)  is analogous, so we omit it.
